Editing Aliasing (section)

== Sampling sinusoidal functions ==
[[File:FFT aliasing 600.gif|thumb|right|300px|Fig.2 '''Upper left:''' Animation depicts a sequence of sinusoids, each with a higher frequency <math display="inline">f</math> than the previous ones. These "true" signals are also being sampled (blue dots) at a constant sampling frequency or rate <math display="inline">f_s</math>.

'''Upper right:''' The ''continuous'' Fourier transform of the sinusoid (not the samples). The single non-zero component, depicting the actual frequency, means that there is no ambiguity.

'''Lower right:''' The ''discrete'' Fourier transform of just the available samples. The presence of two components means that the samples can fit at least two different sinusoids, one of which is with the true frequency (upper-right). Another sinusoid is with an alias frequency <math display="inline">f-f_{\rm s}</math>. (Here the absolute value of it is shown.)

'''Lower left:''' Using the same samples (now in orange), the default reconstruction algorithm produces the lower-frequency sinusoid.]]

[[Sine wave|Sinusoid]]s are an important type of periodic function, because realistic signals are often modeled as the summation of many sinusoids of different frequencies and different amplitudes (for example, with a [[Fourier series]] or [[Fourier transform|transform]]). Understanding what aliasing does to the individual sinusoids is useful in understanding what happens to their sum.

When sampling a function at frequency {{math|''f''{{sub|s}}}} (i.e., the sampling interval is {{math|1/''f''{{sub|s}}}}), the following functions of time {{math|(''t'')}} yield identical sets of samples if the sampling starts from <math display="inline">t=0</math> such that <math>t=\frac{1}{f_s}n</math> where <math display="inline">n = 0,1,2,3</math>, and so on:

<math display="block">\{ \sin(2\pi(f+Nf_s)t+\varphi), N=0,\pm1,\pm2,\pm3, \ldots \}.</math>

A [[frequency spectrum]] of the samples produces equally strong responses at all those frequencies. Without collateral information, the frequency of the original function is ambiguous. So, the functions and their frequencies are said to be ''aliases'' of each other.  Noting the sine functions as odd functions''':'''

:<math>
\sin(2\pi (f+Nf_{\rm s})t + \phi) = \left\{ \begin{array}{ll}
 +\sin(2\pi (f+Nf_{\rm s})t + \phi),
 & f+Nf_{\rm s} \ge 0 \\ 
 -\sin(2\pi |f+Nf_{\rm s}|t - \phi),
 & f+Nf_{\rm s} < 0 \\ 
\end{array} \right.
</math>

thus, we can write all the alias frequencies as positive values: <math>f_{_N}(f) \triangleq \left|f+Nf_{\rm s}\right|</math>. For example, a snapshot of the lower right frame of Fig.2 shows a component at the actual frequency <math>f</math> and another component at alias <math>f_{_{-1}}(f)</math>. As <math>f</math> increases during the animation, <math>f_{_{-1}}(f)</math> decreases. The point at which they are equal <math>(f=f_s/2)</math> is an axis of symmetry called the '''''folding frequency''''', also known as '''''[[Nyquist frequency]]'''''.

Aliasing matters when one attempts to reconstruct the original waveform from its samples. The most common reconstruction technique produces the smallest of the <math>f_{_N}(f)</math> frequencies. So, it is usually important that <math>f_0(f)</math> be the unique minimum. A necessary and sufficient condition for that is <math>f_s/2 > |f|,</math> called the '''''Nyquist condition'''''. The lower left frame of Fig.2 depicts the typical reconstruction result of the available samples. Until <math>f</math> exceeds the Nyquist frequency, the reconstruction matches the actual waveform (upper left frame). After that, it is the low frequency alias of the upper frame.

=== Folding ===
The figures below offer additional depictions of aliasing, due to sampling.  A graph of amplitude vs frequency (not time) for a single sinusoid at frequency {{math|0.6 ''f''{{sub|s}}}} and some of its aliases at {{math|0.4 ''f''{{sub|s}},}} {{math|1.4 ''f''{{sub|s}},}} and {{math|1.6 ''f''{{sub|s}}}} would look like the 4 black dots in Fig.3. The red lines depict the paths ([[wikt:loci|loci]]) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between {{math|''f''{{sub|s}}/2}} and {{math|''f''{{sub|s}}}}). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and {{math|''f''{{sub|s}}.}} Folding is often observed in practice when viewing the [[Frequency spectrum#Spectrum analysis|frequency spectrum]] of real-valued samples, such as Fig.4.
{|
|[[File:Aliasing-folding-2.svg|thumb|x180px|Fig.3: The black dots are aliases of each other. The solid red line is an <u>example</u> of amplitude varying with frequency. The dashed red lines are the corresponding paths of the aliases.]]
|[[File:Example of spectral "folding" caused by sampling a real-valued waveform.png|thumb|x180px|Fig.4: The Fourier transform of music sampled at 44,100 samples/sec exhibits symmetry (called "folding") around the Nyquist frequency (22,050&nbsp;Hz).]]
|[[File:Aliasing-folding.svg|thumb|x180px|Fig.5: Graph of frequency aliasing, showing folding frequency and periodicity.  Frequencies above {{math|''f''{{sub|s}}/2}} have an ''alias'' below {{math|''f''{{sub|s}}/2,}} whose value is given by this graph.]]
|}

[[File:Aliasing between a positive and a negative frequency.svg|thumb|300px|right|Two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points when sampled at the rate ({{math|''f''{{sub|s}}}}) indicated by the grid lines. The case shown here is:  {{math|''f''{{sub|cyan}} {{=}} ''f''{{sub|{{sub|−1}}}}(''f''{{sub|gold}}) {{=}} ''f''{{sub|gold}} – ''f''{{sub|s}}}}]]

=== Complex sinusoids ===

[[Negative frequency#Complex sinusoids|Complex sinusoids]] are waveforms whose samples are [[complex numbers]] (<math display="inline">z = Ae ^{i\theta } = A(\cos \theta + i\sin \theta) </math>), and the concept of [[negative frequency]] is necessary to distinguish them. In that case, the frequencies of the aliases are given by just''':''' {{math|''f''{{sub|{{sub|N}}}}('' f '') {{=}} ''f'' + ''N f''{{sub|s}}.}} (In real sinusoids, as shown in the above, all alias frequencies can be written as positive frequencies <math>f_{_N}(f) \triangleq \left|f+Nf_{\rm s}\right|</math> because of sine functions as odd functions.) Therefore, as {{mvar|f}} increases from {{math|0}} to {{math|''f''{{sub|s}},}} {{math|''f''{{sub|{{sub|−1}}}}('' f '')}} also increases (from {{math|–''f''{{sub|s}}}} to 0). Consequently, complex sinusoids do not exhibit ''folding''.

=== Sample frequency ===

[[File:Aliasing.gif|thumb|300px|Illustration of 4 waveforms reconstructed from samples taken at six different rates. Two of the waveforms are sufficiently sampled to avoid aliasing at all six rates. The other two illustrate increasing distortion (aliasing) at the lower rates.]]

When the condition {{math|''f''{{sub|s}}/2 > '' f ''}} is met for the highest frequency component of the original signal, then it is met for all the frequency components, a condition called the [[Nyquist–Shannon sampling theorem|Nyquist criterion]]. That is typically approximated by filtering the original signal to attenuate high frequency components before it is sampled. These attenuated high frequency components still generate low-frequency aliases, but typically at low enough amplitudes that they do not cause problems. A filter chosen in anticipation of a certain sample frequency is called an [[anti-aliasing filter]].

The filtered signal can subsequently be reconstructed, by interpolation algorithms, without significant additional distortion.  Most sampled signals are not simply stored and reconstructed.  But the fidelity of a theoretical reconstruction (via the [[Whittaker–Shannon interpolation formula]]) is a customary measure of the effectiveness of sampling.